## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 8, Number 4 (1998), 1251-1269.

### No-feedback card guessing for dovetail shuffles

#### Abstract

We consider the following problem. A deck of $2n$ cards labeled
consecutively from 1 on top to $2n$ on bottom is face down on the table. The
deck is given *k* dovetail shuffles and placed back on the table, face
down. A guesser tries to guess at the cards one at a time, starting from top.
The identity of the card guessed at is not revealed, nor is the guesser told
whether a particular guess was correct or not. The goal is to maximize the
number of correct guesses. We show that, for $k \geq 2 \log_2 (2n) + 1$, the
best strategy is to guess card 1 for the first half of the deck and card $2n$
for the second half. This result can be interpreted as indicating that it
suffices to perform the order of $\log_2(2n)$ shuffles to obtain a well-mixed
deck, a fact proved by Bayer and Diaconis. We also show that if $k = c \log_2
(2n)$ with $1 < c < 2$, then the above guessing strategy is not the
best.

#### Article information

**Source**

Ann. Appl. Probab., Volume 8, Number 4 (1998), 1251-1269.

**Dates**

First available in Project Euclid: 9 August 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1028903379

**Digital Object Identifier**

doi:10.1214/aoap/1028903379

**Mathematical Reviews number (MathSciNet)**

MR1661184

**Zentralblatt MATH identifier**

0945.60002

**Subjects**

Primary: 60C05: Combinatorial probability 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Card guessing dovetail shuffle riffle shuffle

#### Citation

Ciucu, Mihai. No-feedback card guessing for dovetail shuffles. Ann. Appl. Probab. 8 (1998), no. 4, 1251--1269. doi:10.1214/aoap/1028903379. https://projecteuclid.org/euclid.aoap/1028903379